ON THE ZEROS OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS. Peter Borwein and Tamás Erdélyi. Abstract. It is proved that a polynomial p of the form

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1 ON THE ZEROS OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS Peter Borwein and Tamás Erdélyi Abstract. It is proved that a polynomial p of the form a j x j, a 0 = 1, a j 1, a j C, has at most c n zeros inside any polygon with vertices on the unit circle, where the constant c depends only on the polygon. Furthermore, if α (0,1) then every polynomial p of the above form has at most c/α zeros inside any polygon with vertices on the circle {z C : z = 1 α}, where the constant c depends only on the number of vertices of the polygon. It is also shown that there is an absolute constant c such that every polynomial of the form a j x j, a 0 = a n = 1, a j 1, a j C, has at most c(nα+ n) zeros in the strip {z C : Im(z) α}, and in the sector {z C : arg(z) α}. These essentially sharp results improve and generalize several earlier results Mathematics Subject Classification. 30C15, 11C08. Key words and phrases. zeros, polynomials, restricted coefficients, small integer coefficients. Research supported in part by the NSERC of Canada (P. Borwein) and by the NSF of the USA under Grant No. DMS (T. Erdélyi) 1 Typeset by AMS-TEX

2 1. Introduction The study of the location of zeros of polynomials from F n := { p : } a i x i, a i { 1,0,1} i=0 begins with Bloch and Pólya [2]. They prove that the average number of real zeros of a polynomial from F n is at most c n. They also prove that a polynomial from F n cannot have more than cnloglogn logn real zeros. This result, which appears to be the first on this subject, shows that polynomials from F n do not behave like unrestricted polynomials. Schur [11] and by different methods Szegő [12] and Erdős and Turán [7] improve the above bound to c nlogn (see also [5]). In [6] we give the right upper bound of c n for the number of real zeros of polynomials from a large class, namely for all polynomials of the form a j x j, a j 1, a 0 = a n = 1, a j C. In this paper we extend this result by proving that a polynomial of the form a j x j, a j 1, a 0 = 1, a j C, cannot have more than c n zeros inside any polygon with vertices on the unit circle, where c depends only on the polygon. We also prove another essentially sharp result stating that a polynomial of the form a j x j, a 0 = a n = 1, a j 1, a j C, has at most c(nα + n) zeros in the strip {z C : Im(z) α}, where c is an absolute constant. Theorems , our main results, have self contained proofs distinct from those in [6]. They sharpen and generalize some results of Amoroso [1], Bombieri and Vaaler [4], and Hua [8], who gave upper bounds for the number of zeros of polynomials with integer coefficients at 1. The class F n and various related classes have been studied from a number of points of view. Littlewood s monograph [9] contains a number of interesting, challenging, and still open problems about polynomials with coefficients from { 1, 1}. The distribution of zeros of polynomials with coefficients from {0, 1} is studied in [10] by Odlyzko and Poonen. 2

3 2. New Results Throughout the paper D(z 0,r) := {z C : z z 0 < r} denotes the open disk of the complex plane centered at z 0 C with radius r > 0. Theorem 2.1. Every polynomial p of the form a j x j, a 0 = 1, a j 1, a j C, has at most c n zeros inside any polygon with vertices on the unit circle, where the constant c depends only on the polygon. Theorem 2.2. There is an absolute constant c such that a j x j, a 0 = a n = 1, a j 1, a j C, has at most c(nα+ n) zeros in the strip and in the sector {z C : Im(z) α}, {z C : arg(z) α}. Theorem 2.3. Let α (0,1). Every polynomial p of the form a j x j, a 0 = 1, a j 1, a j C, has at most c/α zeros inside any polygon with vertices on the circle {z C : z = 1 α}, where the constant c depends only on the number of the vertices of the polygon. The sharpness of Theorem 2.1 can be seen by the theorem below proved in [6]. Theorem 2.A. For every n N, there exists a polynomial p n of the form given in Theorem 2.1 with real coefficients so that p n has a zero at 1 with multiplicity at least n 1. When 0 < α n 1/2, the sharpness of Theorem 2.2 is shown by the polynomials q n (z) := p n (z)+z 2n+1 p n (z 1 ), where p n are the polynomials in Theorem 2.A. Namely the polynomials q n are of the required form with n 1 c(nα+ n) zeros at 1. When n 1/2 α 1, the sharpness of Theorem 2.2 is shown by the polynomials q n (z) := z n 1. The next theorem proved in [3] shows the sharpness of Theorem

4 Theorem 2.B. For every α (0,1), there exists a polynomial p n of the form given in Theorem 2.3 with real coefficients so that p n has a zero at 1 α with multiplicity at least 1/α 1. (It can also be arranged that n 1/α 2 +2.) As a remark to Theorem 2.3 we point out that a more or less straightforward application of Jensen s formula gives that there is an absolute constant c > 0 such that every polynomial p of the form a j x j, a 0 = 1, a j 1, a j C, has at most (c/α)log(1/α) zeros in the disk D(0,1 α), α (0,1). (An interested reader may view this remark as an exercise. We will not use it, and hence will not present its proof.) A very recent (unpublished) example, suggested by Nazarov, shows that this upper bound for the number of zeros in the disk D(0,1 α) is, up to the absolute constant c > 0, best possible. So, in particular, the constant in Theorem 2.3 cannot be made independent of the number of vertices of the polygon. 3. Lemmas To prove our theorems we need some lemmas. Our first lemma states Jensen s formula. Its proof may be found in most of the complex analysis textbooks. Lemma 3.1. Suppose h is a nonnegative integer and f(z) = c k (z z 0 ) k, c h 0 k=h is analytic on the closure of the disk D(z 0,r) and suppose that the zeros of f in D(z 0,r) \ {z 0 } are a 1,a 2,...,a m, where each zero is listed as many times as its multiplicity. Then log c h +hlogr+ m r log a k z 0 = 1 2π k=1 2π 0 log f(z 0 +re iθ ) dθ. A straightforward calculation gives the next lemma. Lemma 3.2. Suppose f is an analytic function on the open unit disk D(0,1) that satisfies the growth condition (1) f(z) 1 1 z, z D(0,1). Let D(z 0,r) D(0,1). Then there is a constant c(r) depending only on r such that 1 2π 2π 0 log f(z 0 +re iθ ) dθ c(r). The next lemma is used in the proof of both Theorems 2.1 and

5 Lemma 3.3. There is an absolute constant c 1 such that every polynomial of the form a j x j, a 0 = 1, a j 1, a j C has at most c 1 (nr+ n) zeros in any open disk D(z 0,r) with z 0 = 1, z 0 C. To prove Lemma 3.3 we need the lemma below. Lemma 3.4 also plays a central role in [6] where its reasonably simple proof may be found. Lemma 3.4. There are absolute constants c 2 > 0 and c 3 > 0 such that ( f(0) c2/a c3 ) exp f [1 a,1], a (0,1], a for every f analytic on the open unit disk D(0, 1) satisfying the growth condition (1) in Lemma 3.2. For the sake of completeness we present the proof of Lemma 3.4 in Section 4. Proof of Lemma 3.3. Without loss of generality we may assume that z 0 := 1 and n 1/2 r 1 (the case 0 < r < n 1/2 follows from the case r = n 1/2, and the case r > 1 is obvious.) Let p be a polynomial of the form given in the lemma. Observe that such a polynomial satisfies the growth condition (1) in Lemma 3.4. Choose a point z 1 [1 r,1] such that ( ) c3 p(z 1 ) exp. r There is such a point by Lemma 3.4. Using the bounds for the coefficients of p, we have log p(z) log((n+1)(1+4r) n ) log(n+1)+4nr, z 1+4r. Let mdenote the numberofzerosofpin the opendisk D(z 1,2r). Applying Jensen s formula on the disk D(z 1,4r), then using the above inequality, we obtain c 3 r +mlog2 log p(z 1) +mlog2 1 2π 2π(log(n+1)+4nr). This, together with n 1/2 r 1, implies m c(nr + n). Now observe that D(1,r) is a subset of D(z 1,2r), and the result follows. Lemma 3.5. Suppose that p is a polynomial of the form a j x j, a 0 = 1, a j 1, a j C Let D(z 0,r) D(0,1). Let r [ 3 4,1). Let 0 < δ < r. Then there is a constant c(r) depending only on r such that p has at most c(r)δ 1 zeros in the open disk D(z 0,r δ). 5

6 Proof. Let p be a polynomial of the form given in the lemma. Observe that r [ 3 4,1) implies that z 0 < 1 4. Hence p(z 0 ) 2 3. Let m denote the number of zeros of p in the open disk D(z 0,r δ). Applying Jensen s formula on D(z 0,r) and Lemma 3.2, we obtain log 2 3 +mlog r r δ log p(z r 0) +mlog r δ 1 2π log p(z 0 +re iθ ) dθ c(r). 2π As 0 < r < 1 and 0 < δ < r, we have log 0 r r δ = log 1 1 (δ/r) δ r δ, and with the previous inequality this implies that m (c(r)+1)δ Proof of Lemma 3.4 In this section, for the sake of completeness, we present the proof of Lemma 3.4 given in [6]. We need some lemmas. Hadamard Three Circles Theorem. Suppose f is regular in For r [r 1,r 2 ], let Then {z C : r 1 z r 2 }. M(r) := max z =r f(z). M(r) log(r2/r1) M(r 1 ) log(r2/r) M(r 2 ) log(r/r1). Corollary 4.1. Let a (0,1]. Suppose f is regular inside and on the ellipse E a with foci at 1 a and 1 a+ 1 4a and with major axis [ 1 a 13a ] 21a,1 a Let Ẽa be the ellipse with foci at 1 a and 1 a+ 1 4a and with major axis [ 1 a 6a ] 14a,1 a Then max f(z) z Ẽa ( max z [1 a,1 a+ 1 4 a] f(z) 6 ) 1/2( ) 1/2 max f(z). z E a

7 Proof. This follows from the Hadamard Three Circles Theorem with the substitution w = a ( ) z +z 1 ( + 1 a+ a ) The Hadamard Three Circles Theorem is applied with r 1 := 1,r := 2, and r 2 := 4. Corollary 4.2. Let Ẽa be as in Corollary 4.1. Then ( ) 1/2 ( 32 max f(z) max 11a f(z) z Ẽa z [1 a,1] ) 1/2 for every f analytic in the open unit disk D(0,1) that satisfies the growth condition (1) (see Lemma 3.2) and for every a (0,1]. Proof. This follows from Corollary 4.1 and the Maximum Principle. Proof of Lemma 3.4. Let h(z) = 1 2 (1 a)(z+z2 ). Observe that h(0) = 0, and there are absolute constants c 4 > 0 and c 5 > 0 such that h(e it ) 1 c 4 t 2, π t π, and for t [ c 5 a,c 5 a], h(e it ) lies inside the ellipse Ẽa. Now let m := 2πc 5 /a +1. Let ξ := exp(2πi/(2m)) be the first 2mth root of unity, and let g(z) := 2m 1 f(h(ξ j z)). Using the Maximum Principle and the properties of h, we obtain ( ) 2 m 1 f(0) 2m = g(0) max g(z) max f(z) z =1 z Ẽa = k=1 ( 1 c 4 (k/m) 2 ( ) ) 2e c max f(z) 6(m 1)( m m 1 4 ( ) 2e c < max f(z) 7(m 1), (m 1)! z Ẽa z Ẽa and the lemma follows by Corollary 4.2. ) 2 5. Proof of the Theorems Proof of Theorem 2.1. It is sufficient to prove the upper bound of the theorem for the number of zeros in a triangle with vertices 0, w, and w 1, where w = 1 and Re(w) 1 2. Let S 1 := D(z 0,r δ), where z 0 := 1 4, r := 3 4, and δ := n 1/2. Let S 2 := D(z 0,r), where z 0 := w and r := cn 1/2. Let S 3 := D(z 0,r), where z 0 := w 1 and r := cn 1/2. Note that if c = c(w) is sufficiently large, then the triangle with vertices 0, w, and w 1 is covered by the union of S 1, S 2, and S 3. Hence the theorem follows from Lemmas 3.3 and

8 Proof of Theorem 2.2. Without loss of generality we may assume that n 1/2 α. it is sufficient to prove that every polynomial p of the form 1 2 a j x j, a 0 = 1, a j 1, a j C, has at most c(nα+ n) zeros in {z C : Im(z) α, z 1}. The remaining part of the theorem follows from this by studying the polynomials q n (z) := z n p n (z 1 ). Let S 1 := D(z 0,r δ), where z 0 := 1 4, r := 3 4, and δ := α. Let S 2 := D(z 0,r δ), where z 0 := 1 4, r := 3 4, and δ := α. Let S 3 := D(z 0,r), where z 0 := 1 and r := 4α. Let S 4 := D(z 0,r), where z 0 := 1 and r := 4α. Note that S 1, S 2, S 3, and S 4 cover {z C : Im(z) α, z 1}. Hence the theorem follows from Lemmas 3.4 and 3.5 (note that n 1/2 α 1 2 implies α 1 nα). Proof of Theorem 2.3. Without loss of generality we may assume that α (0, 1 2 ], otherwise the statement of the theorem is trivial. It is sufficient to prove the upper bound of the theorem for the number of zeros in a triangle with vertices 0, w, and w, where w = 1 α and Re(w) 1 2. Let w =: w eiθ. Let S 1 := D(z 0,r δ), where z 0 := 1 4 eiθ, r := 3 4, δ := α. Let S 2 := D(z 0,r δ), where z 0 := 1 4 e iθ, r := 3 4, δ := α. Note that the triangle with vertices 0, w, and w is covered by the union of S 1 and S 2. Hence the theorem follows from Lemma 3.5. References 1. F. Amoroso, Sur le diamètre transfini entier d un intervalle réel, Ann. Inst. Fourier, Grenoble 40 (1990), A. Bloch and G. Pólya, On the roots of certain algebraic equations, Proc. London Math. Soc. 33 (1932), F. Beaucoup, P. Borwein, D. Boyd, and C. Pinner, Multiple roots of [-1,1] power series, J. London Math. Soc. (to appear). 4. E. Bombieri and J. Vaaler, Polynomials with low height and prescribed vanishing, in Analytic Number Theory and Diophantine Problems, Birkhauser (1987), P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on [0, 1], manuscript. 7. P. Erdős and P. Turán, On the distribution of roots of polynomials, Annals of Math. 57 (1950),

9 8. L.K. Hua, Introduction to Number Theory, Springer-Verlag, Berlin Heidelberg, New York, J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, Lexington, Massachusetts, A. Odlyzko and B. Poonen, Zeros of polynomials with 0,1 coefficients, Ens. Math. 39 (1993), I. Schur, Untersuchungen über algebraische Gleichungen., Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1933), G. Szegő, Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen., Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 (P. Borwein) Department of Mathematics, Texas A&M University, College Station, Texas USA (T. Erdélyi) 9

EXTENSIONS OF THE BLOCH PÓLYA THEOREM ON THE NUMBER OF REAL ZEROS OF POLYNOMIALS

EXTENSIONS OF THE BLOCH PÓLYA THEOREM ON THE NUMBER OF REAL ZEROS OF POLYNOMIALS Tamás Erdélyi Abstract. We prove that there are absolute constants c 1 > 0 and c 2 > 0 for every {a 0, a 1,..., a n } [1,